\( f\left(R\right) \)  theories can be written as an Einstein + massive scalar system. The \( f\left(R\right) \)  action without matter, for example transforms as [1]}, [2]}, [3]}, [4]}, [5]}
\(S &=& \int _{\mathcal {M}}\,d^4x\,\sqrt{-g}\,\left(\frac{\tilde{R}}{16\,\pi }\,+\,\frac{1}{2}\,\tilde{g}^{\mu \nu }\,\phi _{,\mu }\,\phi _{,\nu }\,-\,V\left(\phi \right)\right)  \\\phi &\equiv & \sqrt{\frac{3}{16\,\pi }}\,\ln \Omega  \\V\left(\phi \right) &\equiv & \frac{1}{16\,\pi }\,\frac{f\,-\,R\,f^{\prime }}{\left(f^{\prime }\right)^2} \\\Omega &\equiv & f^{\prime }\) 
